The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness

We provide two alternate settings for the family of varieties, introduced in Huisgen-Zimmermann (J. Pure Appl. Algebra 127 (1998) 39–72), which model the uniserial representations of a finite-dimensional algebra. The first is a quasi-projective subvariety of a Grassmannian containing the members of the mentioned family as a principal affine open cover; among other benefits, one derives invariance from this intrinsic description of the ‘uniserial varieties’. The second viewpoint re-interprets these varieties as locally closed subsets of the traditional module varieties; in particular, it exhibits closedness of the fibres of the canonical maps from the uniserial varieties to the uniserial representations.